Abstract
Very recently, Samet et al. and Jleli and Samet reported that most of fixed point results in the context of G-metric space, defined by Sims and Zead, can be derived from the usual fixed point theorems on the usual metric space. In this paper, we state and prove some fixed point theorems in the framework of G-metric space that cannot be obtained from the existence results in the context of associated metric space.
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1 Introduction and preliminaries
In 2007, Mustafa and Sims introduced the notion of G-metric and investigated the topology of such spaces. The authors also characterized some celebrated fixed point results in the context of G-metric space. Following this initial paper, a number of authors have published many fixed point results on the setting of G-metric space (see, e.g., [1–33] and the references therein). Samet et al. [24] and Jleli and Samet [25] reported that some published results can be considered as a straight consequence of the existence theorem in the setting of the usual metric space. More precisely, the authors of these two papers noticed that is a quasi-metric whenever is a G-metric. It is evident that each quasi-metric induces a metric. In particular, if the pair is a quasi-metric space, then the function defined by
forms a metric on X.
The object of this paper is to get some fixed point results in the context of G-metric space that cannot be concluded from the existence results. This paper can be considered as a continuation of [27], which was inspired by [26].
First, we recollect some necessary definitions and results in this direction. The notion of G-metric spaces is defined as follows.
Definition 1.1 (See [1])
Let X be a non-empty set, be a function satisfying the following properties:
-
(G1)
if ,
-
(G2)
for all with ,
-
(G3)
for all with ,
-
(G4)
(symmetry in all three variables),
-
(G5)
(rectangle inequality) for all .
Then function G is called a generalized metric or, more specifically, a G-metric on X, and the pair is called a G-metric space.
Note that every G-metric on X induces a metric on X defined by
For a better understanding of the subject, we give the following examples of G-metrics.
Example 1.1 Let be a metric space. Function , defined by
for all , is a G-metric on X.
Example 1.2 (See, e.g., [1])
Let . Function , defined by
for all , is a G-metric on X.
In their initial paper, Mustafa and Sims [1] also defined the basic topological concepts in G-metric spaces as follows.
Definition 1.2 (See [1])
Let be a G-metric space, and let be a sequence of points of X. We say that is G-convergent to if
that is, for any , there exists such that for all . We call x the limit of the sequence and write or .
Proposition 1.1 (See [1])
Let be a G-metric space. The following are equivalent:
-
(1)
is G-convergent to x,
-
(2)
as ,
-
(3)
as ,
-
(4)
as .
Definition 1.3 (See [1])
Let be a G-metric space. Sequence is called a G-Cauchy sequence if, for any , there exists such that for all , that is, as .
Proposition 1.2 (See [1])
Let be a G-metric space. Then the following are equivalent:
-
(1)
sequence is G-Cauchy,
-
(2)
for any , there exists such that for all .
Definition 1.4 (See [1])
A G-metric space is called G-complete if every G-Cauchy sequence is G-convergent in .
Definition 1.5 Let be a G-metric space. Mapping is said to be continuous if for any three G-convergent sequences , and converging to x, y and z, respectively, is G-convergent to .
Mustafa [4] extended the well-known Banach [34] contraction principle mapping in the framework of G-metric spaces as follows.
Theorem 1.1 (See [4])
Let be a complete G-metric space and be a mapping satisfying the following condition for all :
where . Then T has a unique fixed point.
Theorem 1.2 (See [4])
Let be a complete G-metric space and be a mapping satisfying the following condition for all :
where . Then T has a unique fixed point.
Remark 1.1 We notice that condition (2) implies condition (3). The converse is true only if . For details see [4].
Lemma 1.1 [4]
By the rectangle inequality (G5) together with the symmetry (G4), we have
2 Main results
Theorem 2.1 Let be a complete G-metric space and be a mapping satisfying the following condition for all :
where . Then T has a unique fixed point.
Proof Let be an arbitrary point, and define the sequence by . By (5), we have
Continuing in the same argument, we will get
Moreover, for all ; , we have by rectangle inequality that
and so, , as . Thus, is G-Cauchy sequence. Due to the completeness of , there exists such that is G-convergent to u.
Suppose that , then
taking the limit as , and using the fact that function G is continuous, then
This contradiction implies that .
To prove uniqueness, suppose that such that , and use Lemma 1.1, then
which implies that . □
Example 2.1 Let and
be a G-metric on X. Define by . Then the condition of Theorem 2.1 holds. In fact,
and
and so,
That is, conditions of Theorem 2.1 hold for this example.
Corollary 2.1 Let be a complete G-metric space and be a mapping satisfying the following condition for all :
where . Then T has a unique fixed point.
Theorem 2.2 Let be a complete G-metric space and be a mapping satisfying the following condition for all , where
Then T has a unique fixed point.
Proof Take . We construct sequence of points in X in the following way:
Notice that if for some , then obviously T has a fixed point. Thus, we suppose that
for all .
That is, we have
From (12), with and , we have
which implies that
and so,
where . Then
for all . Note that from (G3), we know that
with , and by Lemma 1.1, we know that
Then by (13), we have
Moreover, for all ; , we have by rectangle inequality that
and so, , as . Thus, is G-Cauchy sequence. Due to the completeness of , there exists such that is G-convergent to z. From (12), with and , we have
Then
Taking limit as in the inequality above, we have
Now, if , then T has a fixed point. Hence, we assume that . Therefore, by (G3), we get
which implies that , i.e., . □
At first, we assume that
and
where if and only if .
Theorem 2.3 Let be a complete G-metric space and be a mapping satisfying the following condition for all , where and holds
Then T has a unique fixed point.
Proof Take . We construct sequence of points in X in the following way:
Notice that if for some , then obviously T has a fixed point. Thus, we suppose that
for all .
By (G2), we have
From (15), with and , we have
which implies that
then . So sequence is a decreasing sequence in , and thus, it is convergent, say . We claim that . Suppose, to the contrary, that . Taking limit as in (16), we get
which implies . That is, , which is a contrary. Hence, , i.e.,
We shall show that is a G-Cauchy sequence. Suppose, to the contrary, that there exists , and sequence of such that
with . Further, corresponding to , we can choose in such a way that it is the smallest integer with satisfying (19). Hence,
By Lemma 1.1 and (G5), we have
where . Letting in (21), we derive that
Also, by Lemma 1.1 and (G5), we obtain the following inequalities:
and
Letting in (23) and (24) and applying (22), we find that
Again, by Lemma 1.1 and (G5), we have
and
Taking limit as in (26) and (27) and applying (25), we have
By (15), with and , we have
Taking limit as in the inequality above and applying, we have
which implies , which is a contradiction. Then
That is, is a Cauchy sequence. Since is a G-complete, then there exist such that as . From (15), with and , we have
Taking limit as , we get
Then , i.e., . To prove uniqueness, suppose that , such that . Now, by (15), we get
which implies that , i.e., . □
If we take and in Theorem 2.3, where , then we deduce the following corollary.
Corollary 2.2 Let be a complete G-metric space and be a mapping satisfying the following condition for all , where holds
Then T has a unique fixed point.
Example 2.2 Let and
be a G-metric on X. Define by . Then all the conditions of Corollary 2.2 (Theorem 2.3) hold. Indeed,
and
and so,
That is, the conditions of Corollary 2.2 (Theorem 2.3) hold for this example.
Corollary 2.3 Let be a complete G-metric space and be a mapping satisfying the following condition for all , where holds
Then T has a unique fixed point.
Proof By taking , we get
where . That is, conditions of Theorem 2.3 hold, and T has a unique fixed point. □
3 Fixed point results for expansive mappings
In this section, we establish some fixed point results for expansive mappings.
Theorem 3.1 Let be a complete G-metric space and be an onto mapping satisfying the following condition for all , where holds
Then T has a unique fixed point.
Proof Let , since T is onto, then there exists such that . By continuing this process, we get for all . In case , for some , then it is clear that is a fixed point of T. Now, assume that for all n. From (30), with and , we have
which implies that
where . Then we have
By Lemma 1.1, we get
Following the lines of the proof of Theorem 2.1, we derive that is a Cauchy sequence. Since is complete, then there exists such that as . Consequently, since T is onto, then there exists such that . From (30), with and , we have
Taking limit as in the inequality above, we get
That is, . Then . To prove uniqueness, suppose that such that and . Now by (30), we get
which is a contradiction. Hence, . □
Theorem 3.2 Let be a complete G-metric space and be a mapping satisfying the following condition for all , where
Then T has a unique fixed point.
Proof Let , since T is onto, then there exists such that . By continuing this process, we get for all . In case , for some , then it is clear that is a fixed point of T. Now, assume that for all n. From (34), with and , we have
which implies that
and so,
where . By the mimic of the proof of Theorem 2.1, we can show that is a Cauchy sequence. Since is a complete G-metric space, then there exists such that as . Consequently, since T is onto, then there exists such that . From (34), with and , we have
Taking limit as in the inequality above, we have . That is, . To prove the uniqueness, suppose that such that and . □
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Asadi, M., Karapınar, E. & Salimi, P. A new approach to G-metric and related fixed point theorems. J Inequal Appl 2013, 454 (2013). https://doi.org/10.1186/1029-242X-2013-454
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DOI: https://doi.org/10.1186/1029-242X-2013-454